Summary: Finite decomposition complexity (FDC) is a large scale property of a metric space. It generalizes finite asymptotic dimension and applies to a wide class of groups. To make the property quantitative, a countable ordinal “the complexity” can be defined for a metric space with FDC. In this paper we prove that the subgroup \(\mathbb Z\wr\mathbb Z\) of Thompson’s group \(F\) belongs to \(\mathcal D_\omega\) exactly, where \(\omega\) is the smallest infinite ordinal number and show that \(F\) equipped with the word-metric with respect to the infinite generating set \(\{x_0,x_1,\dots,x_n,\dots\}\) does not have finite decomposition complexity.